Amplitude instability, phase transitions, and dynamic properties in two-dimensional Yukawa clusters

The conditions for the amplitude instability onset in a two-dimensional Yukawa cluster system consisting of seven particles are investigated for the case of sufficiently large thermal particle displacements. We consider here an analytical approach to derive the criterion for such instability onset, which is based on searching an energetically favorable inflection point of the system potential energy. We show that there is a proportionality between melting curve for an “infinite” two-dimensional system with the hexagonal lattice and analytical curve of critical points for the seven-particle Yukawa cluster. We use numerical simulations to study the structural and dynamical properties of the system near the critical point of the amplitude instability. The obtained results are compared to the criteria for melting of the “infinite” two-dimensional system.

Figure 1

Illustration of particle positions in a two-dimensional hexagonal cluster. Directions of particle displacements from ideal positions for the system close to the amplitude instability onset are indicated by arrows.

Figure 2

Normalized value of the electric-field gradient ${\alpha }^{*}$ vs the screening parameter κ.

Figure 3

Plot of $U^{*}$, ${U^{*}}^{\prime }$, and ${U^{*}}^{″}$ vs $\delta y/l_o$; the cross sections of $U^{*}(\delta x,\delta y)$, ${U^{*}}^{\prime }(\delta x,\delta y)$, and ${U^{*}}^{″}(\delta x,\delta y)$ are displayed along the following lines: (a) $\delta x\cong 2.4\delta y$ for $\kappa =0$ and (b) $\delta x\cong 1.8\delta y$ for $\kappa =4$. The dashed lines denote the inflection point ($\delta x_{\mathrm{IP}},\delta y_{\mathrm{IP}}$).

Figure 4

Dependencies ${\mathrm{\Gamma }}_c\left(\kappa \right)$ and ${\mathrm{\Gamma }}_m\left(\kappa \right)$ defined in Eqs. (4) and (1) accordingly (solid lines). Symbols ($•,◯$) denote in situ experimental observation of phase transitions in the seven-particle cluster; ${\mathrm{\Gamma }}_o\approx 76$ ($•$) corresponds to the quasicrystal state and ${\mathrm{\Gamma }}_o\approx 25$ ($◯$) corresponds to the liquid state [28].

Figure 5

Dependence of the ratio ${\mathrm{\Gamma }}_m/{\mathrm{\Gamma }}_c$ on the screening parameter κ. The gray area shows a 5% absolute deviation from the dashed line ${\mathrm{\Gamma }}_m/{\mathrm{\Gamma }}_c=11.25$.

Figure 6

Dependence of the normalized radial component of particle displacement $\delta r_{\mathrm{IP}}/l_o$ (averaged over the ensemble) as a function of the screening parameter κ. The gray area shows a 5% absolute deviation from the dashed line, $\delta r_{\mathrm{IP}}/l_o\cong 0.22$.

Figure 7

The pair-correlation functions $g\left(l/l_o\right)$ for $\kappa =0$ and for different values of ${\mathrm{\Gamma }}_o=Q^2/T{l}_o$.

Figure 8

(a) The normalized positions of the first peak of $g\left(l/l_o\right),l_{dm}/l_o$ and (b) the normalized height of the first peak of $g\left(l/l_o\right)$, ${g_m}^{*}=g_m/\sqrt{{\mathrm{\Gamma }}_o}$, as functions of ${\mathrm{\Gamma }}_o/{\mathrm{\Gamma }}_c$ for two different values of the screening parameter.

Figure 9

Particle thermal motion trajectories for $\kappa =0$, $\xi \cong 1$, for different ${\mathrm{\Gamma }}_o$ and time series Δt.

Figure 10

Dependences (a) $R^{2*}=\langle {\left(\delta r\right)}^2\rangle /2{d_o}^2$ and (b) $D^{*}=R^{2*}{\mathrm{\Gamma }}_o/\nu t$ vs $\nu t$ for $\kappa =3\left({\mathrm{\Gamma }}_c\cong 18.9\right)$; within the region νt < 100 the coupling parameter values ${\mathrm{\Gamma }}_o$ are smaller for greater values of $R^{2*}$. (The gray curves correspond to the region below the threshold, and black curves correspond to the region above the threshold.)

Figure 11

Dependencies of $B_c$ (white symbols) and $B_l$ (gray symbols) on ${\mathrm{\Gamma }}_o/{\mathrm{\Gamma }}_c$.